3D Adaptive VEM with stabilization-free a posteriori error bounds

TL;DR

This paper extends adaptive virtual element methods to 3D tetrahedral meshes, providing stabilization-free error bounds and demonstrating improved mesh refinement efficiency over standard methods.

Contribution

It introduces a stabilization-free a posteriori error control for 3D AVEMs, enabling more efficient mesh refinement strategies in three-dimensional problems.

Findings

Reduces the number of refined cells by about 30% for a given error threshold.

Establishes residual-type error bounds that eliminate the need for stabilization terms.

Demonstrates effectiveness on Fichera corner domain with numerical tests.

Abstract

The present paper extends the theory of Adaptive Virtual Element Methods (AVEMs) to the three-dimensional meshes showing the possibility to bound the stabilization term by the residual-type error estimator. This new bound enables a stabilization-free a posteriori control for the energy error. Following the recent studies for the bi-dimensional case, we investigate the case of tetrahedral elements with aligned edges and faces. We believe that the AVEMs can be an efficient strategy to address the mesh conforming requirements of standard three-dimensional Adaptive Finite Element Methods (AFEMs), which typically extend the refinement procedure to non-marked mesh cells. Indeed, numerical tests on the Fichera corner shape domain show that this method can reduce the number of three-dimensional cells generated in the refinement process by about 30% with compared to standard AFEMs, for a given error threshold.